The chromatic number of almost stable Kneser hypergraphs

Let $V(n,k,s)$ be the set of $k$-subsets $S$ of $[n]$ such that for all $i,j\in S$, we have $|i-j|\geq s$ We define almost $s$-stable Kneser hypergraph $KG^r{{[n]}\choose k}_{s{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ to be the $r$-uniform hypergraph whose vertex set is $V(n,k,s)$ and whose edges are the $r$-uples of disjoint elements of $V(n,k,s)$. With the help of a $Z_p$-Tucker lemma, we prove that, for $p$ prime and for any $n\geq kp$, the chromatic number of almost 2-stable Kneser hypergraphs $KG^p {{[n]}\choose k}_{2{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ is equal to the chromatic number of the usual Kneser hypergraphs $KG^p{{[n]}\choose k}$, namely that it is equal to $\lceil\frac{n-(k-1)p}{p-1}\rceil.$ Defining $\mu(r)$ to be the number of prime divisors of $r$, counted with multiplicities, this result implies that the chromatic number of almost $2^{\mu(r)}$-stable Kneser hypergraphs $KG^r{{[n]}\choose k}_{2^{\mu(r)}{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ is equal to the chromatic number of the usual Kneser hypergraphs $KG^r{{[n]}\choose k}$ for any $n\geq kr$, namely that it is equal to $\lceil\frac{n-(k-1)r}{r-1}\rceil.

Fan's lemma via bistellar moves

Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free Z2-action, and use it to give a combinatorial proof of Fan's lemma about labellings of centrally symmetric triangulations of spheres. Similarly to other combinatorial proofs, we must assume an additional property of the triangulation for the proof to work. However, unlike the other combinatorial proofs, no such assumption is needed for dimensions at most 3

Combinatorial Stokes formulae

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